Related papers: Asymmetric integrable quad-graph equations
A new method for the Lie group classification of differential equations is proposed. It is based of the determination of all possible cases of linear dependence of certain indeterminate appearing in the determining equations of symmetries…
Integrable discrete scalar equations defined on a~two or a three dimensional lattice can be rewritten as difference systems in bond variables or in face variables respectively. Both the difference systems in bond variables and the…
This paper presents a systematic investigation of the integrability conditions for nonautonomous quad-graph maps, using the Lax pair approach, the ultra-local singularity confinement criterion and direct construction of conservation laws.…
We analyze the factorization process for lattice maps, searching for integrable cases. The maps were assumed to be at most quadratic in the dependent variables, and we required minimal factorization (one linear factor) after 2 steps of…
A method of classification of integrable equations on quad-graphs is discussed based on algebraic ideas. We assign a Lie ring to the equation and study the function describing the dimensions of linear spaces spanned by multiple commutators…
A fifth-order KdV equation with time dependent coefficients and linear damping has been studied. Symmetry groups have several different applications in the context of nonlinear differential equations. For instance, they can be used to…
We suggest an approach for description of integrable cases of the Abel equations. It is based on increasing of the order of equations up to the second one and using equivalence transformations for the corresponding second-order ordinary…
Nonlinear self-adjointness method for constructing conservation laws of partial differential equations (PDEs) is further studied. We show that any adjoint symmetry of PDEs is a differential substitution of nonlinear self-adjointness and…
We describe a method to construct well-posed initial value problems for not necessarily integrable equations on not necessarily simply connected quad-graphs. Although the method does not always provide a well-posed initial value problem…
We derive a method for finding Lie Symmetries for third-order difference equations. We use these symmetries to reduce the order of the difference equations and hence obtain the solutions of some third-order difference equations. We also…
Aspects of the algebraic structure and representation theory of the quantum affine superalgebras with symmetrizable Cartan matrices are studied. The irreducible integrable highest weight representations are classified, and shown to be…
We show that the so-called hidden potential symmetries considered in a recent paper [Gandarias M., Physica A, 2008, V.387, 2234-2242] are ordinary potential symmetries that can be obtained using the method introduced by Bluman and…
Integrability conditions for difference equations admitting a second order formal recursion operator are presented and the derivation of symmetries and canonical conservation laws is discussed. In the generic case, nonlocal conservation…
A class of partial differential equations (a conservation law and four balance laws), with four independent variables and involving sixteen arbitrary continuously differentiable functions, is considered in the framework of equivalence…
In this paper we consider the classification of dispersive linearizable partial difference equations defined on a quad-graph by the multiple scale reduction around their harmonic solution. We show that the A_1, A_2 and A_3 linearizability…
We present a method to obtain symmetries for second-order systems of ordinary difference equations and how to use them to reduce the order. We also introduce a technique of finding conservation laws for such systems.
For partial differential equations (PDEs) that have $n\geq2$ independent variables and a symmetry algebra of dimension at least $n-1$, an explicit algorithmic method is presented for finding all symmetry-invariant conservation laws that…
Let $\tau\colon\tilde{\mathcal{E}}\to\mathcal{E}$ be a differential covering of a PDE $\tilde{\mathcal{E}}$ over $\mathcal{E}$. We prove that if $\mathcal{E}$ possesses infinite number of symmetries and/or conservation laws then…
The notions of generating sets of conservation laws of systems of differential equations with respect to symmetry groups and equivalence groups are introduced and applied. This allows us to generalize essentially the procedure of finding…
We address the problem of classification of integrable differential-difference equations in 2+1 dimensions with one/two discrete variables. Our approach is based on the method of hydrodynamic reductions and its generalisation to dispersive…